Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $z \neq 0$. $a = \dfrac{z^2 + 2z - 15}{z + 10} \div \dfrac{-6z - 30}{z + 10} $
Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{z^2 + 2z - 15}{z + 10} \times \dfrac{z + 10}{-6z - 30} $ First factor the quadratic. $a = \dfrac{(z + 5)(z - 3)}{z + 10} \times \dfrac{z + 10}{-6z - 30} $ Then factor out any other terms. $a = \dfrac{(z + 5)(z - 3)}{z + 10} \times \dfrac{z + 10}{-6(z + 5)} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac{ (z + 5)(z - 3) \times (z + 10) } { (z + 10) \times -6(z + 5) } $ $a = \dfrac{ (z + 5)(z - 3)(z + 10)}{ -6(z + 10)(z + 5)} $ Notice that $(z + 10)$ and $(z + 5)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac{ \cancel{(z + 5)}(z - 3)(z + 10)}{ -6(z + 10)\cancel{(z + 5)}} $ We are dividing by $z + 5$ , so $z + 5 \neq 0$ Therefore, $z \neq -5$ $a = \dfrac{ \cancel{(z + 5)}(z - 3)\cancel{(z + 10)}}{ -6\cancel{(z + 10)}\cancel{(z + 5)}} $ We are dividing by $z + 10$ , so $z + 10 \neq 0$ Therefore, $z \neq -10$ $a = \dfrac{z - 3}{-6} $ $a = \dfrac{-(z - 3)}{6} ; \space z \neq -5 ; \space z \neq -10 $